Given a matrix $A_1$ as part of the equation $A\vec{x}=\vec{b}$:
$$
\begin{bmatrix}
P & f & f & f\\
0 & P & f & f\\
0 & 0 & P & f
\end{bmatrix}
$$
What do we know based on the fact that there is a pivot in every row?
Given a matrix $A_2$ as part of the equation $A\vec{x}=\vec{b}$:
$$
\begin{bmatrix}
P & f & f\\
0 & P & f\\
0 & 0 & P\\
0 & 0 & 0
\end{bmatrix}
$$
What do we know based on the fact that there is a pivot in every column?
My understanding is that a pivot in every row (as in $A_1$) tells us that the columns of $A_1$ span $\mathbb{R}^m$. And that a pivot in every column (as in $A_2$) tells us that the columns are linearly independent. Are these understandings correct?
I'm sure that we know a lot about a matrix given the conditions listed above, but I'm just looking for the most obvious or helpful information.
Best Answer
A pivot in every row means that the linear system $Ax=b$ has at least one solution, for every $b$.
If every column has a pivot, then the linear system $Ax=b$ has at most one solution.
If both hold (which can happen only if $A$ is a square matrix), we get that the system $Ax=b$ has unique solution for every $b$.
A pivot in every row is equivalent to $A$ having a right inverse, and equivalent to the columns of $A$ spanning $\mathbb{R}^m$ ($m$ is the number of rows).
A pivot in every column is equivalent to $A$ having a left inverse, and equivalent to the columns of $A$ being linearly independent.